Book Description
Lie algebras have many varied applications, both in mathematics and mathematical physics. This book provides a thorough but relaxed mathematical treatment of the subject, including both the Cartan-Killing-Weyl theory of finite dimensional simple algebras and the more modern theory of Kac-Moody algebras. Proofs are given in detail and the only prerequisite is a sound knowledge of linear algebra. The Appendix provides a summary of the basic properties of each Lie algebra of finite and affine type.Download Description
Lie algebras have many varied applications, both in mathematics and mathematical physics. This book provides a thorough but relaxed mathematical treatment of the subject, including both the Cartan-Killing-Weyl theory of finite dimensional simple algebras and the more modern theory of Kac-Moody algebras. Proofs are given in detail and the only prerequisite is a sound knowledge of linear algebra. The first half of the book deals with classification of the finite dimensional simple Lie algebras and of their finite dimensional irreducible representations. The second half introduces the theory of Kac-Moody algebras, concentrating particularly on those of affine type. A brief account of Borcherds algebras is also included. An Appendix gives a summary of the basic properties of each Lie algebra of finite and affine type. ... Read more

Book Description

Lie algebras are efficient tools for analyzing the properties of physical systems. Concrete applications comprise the formulation of symmetries of Hamiltonian systems, the description of atomic, molecular and nuclear spectra, the physics of elementary particles and many others. This work gives an introduction to the properties and the structure of the Lie algebras su(n). First, characteristic quantities such as structure constants, the Killing form and functions of Lie algebras are introduced. The properties of the algebras su(2), su(3) and su(4) are investigated in detail. Geometric models of the representations are developed. A lot of care is taken over the use of the term "multiplet of an algebra".
The book features an elementary (matrix) access to su(N)-algebras, and gives a first insight into Lie algebras. Student readers should be enabled to begin studies on physical su(N)-applications, instructors will profit from the detailed calculations and examples.

Customer Reviews (4)

Really about time su(n) representaions like these were available
For me the su(2),su(3) material wasn't new.For me the su(2),su(3) material wasn't new.
I was grateful for the representation of su(4) and the structure constants.
The expansion of the Gell-Mann matrices is consistent with standard physic text
notation ( so the su(4) structure constants in Gordon Kane's "Modern Elementary Particle Physics" agree exactly).
The explanation of Young tableau is welcome andthe subgroup structures given
illuminate the Dynkin diagrams for these Lie algebras as well.
I find myself wishing this text were available in the 70's or 80's.
The exposition is clear and covers the material well.
One needs some modern algebra background and a familiarity with matrix representation notation.His explanation of how two so(n) groups are Hermitian in su(n) is something it took me years to figure out on my own!
I say this is a book well done and one that has been needed for a long time.
I would that he had expanded the book with an su(5) representations, structure constants
and sub-algebras
I was grateful for the representaion of su(4) and the structure constants.
The expansion of the Gell-Mann matrices is consistent with standard physic text notation ( so the su(4) structure constants in Gordon Kane's "Modern Elementry Particle Physics" agree exactly).
The explaination of Young tableau is welcome andthe subgroup structures givenilluninate the Dynkin diagrams for these Lie algebras as well.
I find myself wishing this text were available in the 70's or 80's.
The exposition is clear and covers the material well.
One needs some modern algebra backgrond and a familarity with matrix representaion notation.
His explaination of how two so(n) groups are Hermetian in su(n) is something it took me years to figure out on my own!
I say this is a book well done and one that has been needed for a long time.I would that he had expanded the book with an su(5) representations, structure constants and sub-algebras

GREAT!
Great indtroductory book, very user friendly.It explains and shows in an easy to understand and simple way....lots of examples and explanations. EASY AND ENJOYABLE READ.

Lie algebra demystified
A practical introduction to an esoteric topic which frightens many physics students.

This book presupposes little background mathematics and begins by defining lie alegebras and providing adequate examples. He then details some basic properties of finite dimensional lie algebras and offers several ways of "representing" them including the adjoint representation. From the beginning there is an emphasis on applications to quantum mechanics and I especially enjoyed the section on SU ( 2) and it's application to angular momentum operators. SU ( 3) and SU ( 4 ) are developed in due time in a logical and easy to understand format.

He also shows, in a simple way, how the tangent space of the identity of a lie group has a lie algebra structure which is useful in studying the group's local properties.

A very handy reference for those studying advanced quantum mechanics and particle physics yet basic enough for undergraduates to grasp the concepts.

An excellent practical guide
This short book covers an important aspect that has been neglected by most textbooks on Lie algebras written for physicists, namely providing a comprehensible introduction for undergraduates based on detailed examples, computations and precise motivations, without having to develop the formal theory. This is not a textbook on Lie algebras in the usual sense, but a practical guide whose intention is to provide a solid comprehension of the main facts on (finite dimensional) Lie algebras used in physics. This justifies the choice of the objects analyzed, the compact real form su(N) of the Lie algebras sl(N,C), which constitute an essential tool in the study of the interacting boson model and nuclear rotational states. The topics covered by this book are quite modest (there are no general proofs and no development of classical problems like the classification of simple Lie algebras), and focuses on a detailed comment on the properties of simple algebras using mainly three Lie algebras, su(2),su(3) and su(4), before ennouncing the general case in the last chapter. However, this should not be understated, specially because the book explains carefully the usual notations (which change in the literature from author to author) and tries to clarify the reasons that justify the study of the formal theory.
The book is divided into six chapters, which we comment separately. The first chapter is a quick and effective overview on the basic properties of simple Lie algebras, namely the adjoint representation, the Killing form, representations and their reducibility. For the inner product the Dirac bracket notation is used. The concept of multiplets, which plays an essential role, is introduced at the end of this chapter. Chapter 2 begins with a short discussion of hermitian matrices, and introduces the Lie algebra su(N) in the usual way. The complexification of this algebra is shortly commented, as well as the generation of the algebra by means of operators. The structure constants over the standard basis are obtained, and as application the Killing form for su(N) is computed. It should be said that the notations used in this chapter have in mind the Gell-Mann matrices, which will be introduced later. Chapter 3 studies the fundamental facts concerning the rank one algebra su(2), and which will be central to later developments. The topics commented are generators of su(2), that is, the Pauli matrices, the quantum mechanical operators J of angular momentum, the su(2) multiplets and the irreducible (complex) representations. Further the tensor products (called "direct products") of these representations and their decomposition into irreducible components is commented. Many very detailed computations are presented, which illustrate clearly the procedure and its significance. Moreover the graphical method for the tensor product decomposition is developed,
The fourth chapter, devoted to the Lie algebra su(3), which cosntitutes in some sense the core of this book, actually develops the main aspects necessary to the description of global symmetry schemes for hadrons (without deeping into the actual classification, for this would require a basic knowledge of quantum field theory). The Lie algebra su(3) is introduced according Gell-Mann's notation. The step operators and states of su(3) are introduced, and the individual states and multiplicities are carefully constructed using graphical motivation (which actually corresponds to the standard application of the su(2)-triples). In order to formalize the construction, the Young tableaux are used (these constituting an essential tool for the analysis of the su(N) algebras). Special attention is devoted to the fundamental su(3)-multiplet (the quark representation 3) and its dual. This leads naturally to the introduction of the hypercharge Y (however no reference to the Gell-Mann-Nishijima formula is made). The (quadratic) Casimir operator of su(3) and its eigenvalues are analyzed, with explicit examples that point out the main properties of this invariant. The next section focuses on the tensor products of su(3)-multiplets, and develops also the graphical method to deduce the decomposition. A table presents some of these tensor products (for highest weights lower or equal to (2,1)). Again, this motivation is used to present the Young tableaux. Chapter 5 presents more or less the same topics for the rank three algebra su(4), and discusses the charm C (as a natural consequence of the quantum numbers discussed for su(3)). The multiplets and tensor products are reviewed (the diagrams are of exceptional quality and clarity), and the chapter finishes commenting on the standard Weyl basis (that is, the basis obtained from the root system of the corresponding algebra; this is the presentation that will be found in almost any book on Lie algebras). These facts are presented without proof, but serve to illustrate fundamental facts like the Cartan integers or the presentation by generators and relations that the interested reader will find in any standard text. Chapter six gives a recopilation of the basic facts of the su(N) algebras for arbitrary values of N (hermitian generators and multiplets, quadratic Casimir operator, etc). The bibliography presents some texts to profound the study. A little remark: the reference to Cornwell's book refers specifically to volume II, which deals with the theory of finite dimensional Lie algebras.
On balance I think this book is an excellent first contact with Lie algebras for those using them in physics, because of the lucid style and the clarity in the exposition. The very detailed calculations and step by step introduction of the material allow the readers not familiar with Lie algebras tobecome confident with the main facts they will find in any standard textbook, and which often discourages because of notational problems or implicit assumption of knowledge concerning the fundamental properties. Although the notation is mainly that used in physics literature, the examples and motivations introduced in this text will help the reader in the transition to other books using alternative notations. This work is a welcome reference for both beginners in Lie algebras for physics, as well as for instructors. ... Read more

Product Description
This textbook comprehensively introduces students and researchers to the application of continuous symmetries and their Lie algebras to ordinary and partial differential equations. Covering all the modern techniques in detail, it relates applications to cutting-edge research fields such as Yang Mills theory and string theory.Aimed at readers in applied mathematics and physics rather than pure mathematics, the material is ideally suited to students and researchers whose main interest lies in finding solutions to differential equations and invariants of maps.A large number of worked examples and challenging exercises help readers to work independently of teachers, and by including SymbolicC++ implementations of the techniques in each chapter, the book takes full advantage of the advancements in algebraic computation.Twelve new sections have been added in this edition, including: Haar measure, Sato's theory and sigma functions, universal algebra, anti-self dual Yang Mills equation, and discrete Painlevé equations. ... Read more

Book Description
From the reviews of the French edition: "This is a rich and useful volume. The material it treats has relevance well beyond the theory of Lie groups and algebras, ranging from the geometry of regular polytopes and paving problems to current work on finite simple groups having a (B,N)-pair structure, or "Tits systems". A historical note provides a survey of the contexts in which groups generated by reflections have arisen. A brief introduction includes almost the only other mention of Lie groups and algebras to be found in the volume. Thus the presentation here is really quite independent of Lie theory. The choice of such an approach makes for an elegant, self-contained treatment of some highly interesting mathematics, which can be read with profit and with relative ease by a very wide circle of readers (and with delight by many, if the reviewer is at all representative)."
(G.B. Seligman in MathReviews) ... Read more

Book Description

Semisimple Lie groups, and their algebraic analogues over fields other than the reals, are of fundamental importance in geometry, analysis, and mathematical physics. Three independent, self-contained volumes, under the general title "Lie Theory," feature survey work and original results by well-established researchers in key areas of semisimple Lie theory.

Book Description

Book Description

This Ergebnisse volume is aimed at a wide readership of mathematicians and physicists, graduate students and professionals. The main thrust of the book is to show how algebraic geometry, Lie theory and Painlevé analysis can be used to explicitly solve integrable differential equations and construct the algebraic tori on which they linearize; at the same time, it is, for the student, a playing ground to applying algebraic geometry and Lie theory. The book is meant to be reasonably self-contained and presents numerous examples. The latter appear throughout the text to illustrate the ideas, and make up the core of the last part of the book. The first part of the book contains the basic tools from Lie groups, algebraic and differential geometry to understand the main topic.

Book Description

This book starts with the elementary theory of Lie groups of matrices and arrives at the definition, elementary properties, and first applications of cohomological induction, which is a recently discovered algebraic construction of group representations. Along the way it develops the computational techniques that are so important in handling Lie groups. The book is based on a one-semester course given at the State University of New York, Stony Brook in fall, 1986 to an audience having little or no background in Lie groups but interested in seeing connections among algebra, geometry, and Lie theory.

These notes develop what is needed beyond a first graduate course in algebra in order to appreciate cohomological induction and to see its first consequences. Along the way one is able to study homological algebra with a significant application in mind; consequently one sees just what results in that subject are fundamental and what results are minor.

Book Description
This volume begins with an introduction to the structure offinite-dimensional simple Lie algebras, including therepresentation of ${\widehat {\mathfrak {sl}}}(2, {\mathbb C})$,root systems, the Cartan matrix, and a Dynkin diagram of afinite-dimensional simple Lie algebra. Continuing on, the mainsubjects of the book are the structure (real and imaginary rootsystems) of and the character formula for Kac-Moodysuperalgebras, which is explained in a very general setting.Only elementary linear algebra and group theory are assumed.Also covered is modular property and asymptotic behavior ofintegrable characters of affine Lie algebras. The exposition isself-contained and includes examples. The book can be used in agraduate-level course on the topic. ... Read more

Book Description
The aim of the present work is two-fold. Firstly it aims at a giving an account of many existing algorithms for calculating with finite-dimensional Lie algebras. Secondly, the book provides an introduction into the theory of finite-dimensional Lie algebras. These two subject areas are intimately related. First of all, the algorithmic perspective often invites a different approach to the theoretical material than the one taken in various other monographs (e.g., [42], [48], [77], [86]). Indeed, on various occasions the knowledge of certain algorithms allows us to obtain a straightforward proof of theoretical results (we mention the proof of the Poincaré-Birkhoff-Witt theorem and the proof of Iwasawa's theorem as examples). Also proofs that contain algorithmic constructions are explicitly formulated as algorithms (an example is the isomorphism theorem for semisimple Lie algebras that constructs an isomorphism in case it exists). Secondly, the algorithms can be used to arrive at a better understanding of the theory. Performing the algorithms in concrete examples, calculating with the concepts involved, really brings the theory of life. ... Read more

Book Description
The first contribution covers the theory of finite groups of Lie type, which is an important field of current mathematical research. After giving the basic information Carter describes the Deligne-Lusztig method of obtaining characters of these groups using l-adic cohomology and subsequent work of Lusztig.The second part by Platonov and Yanchevskii surveys the structure of finite-dimensional division algebras and includes an account of reduced K-theory. ... Read more

Book Description
There is a fruitful and fascinating interaction between infinite dimensional operator theory (particularly decomposable, scalar and spectral generalized operator theory due to C. Foias and I. Colojoara) and Lie algebra theory. The present book is the first devoted to this field, ranging from some short historical notes to the most recent developments. Nilpotence criteria, infinite dimensional variants of Lie's theorem for solvable systems of bounded operators, spectral properties of elements of semisimple Lie algebras and simultaneous triangularisation are expounded. The book is self-contained and features an extensive bibliography. It is aimed at postgraduate students and researchers who are introduced to an interesting recent area of research and will learn some new methods useful for both of the domains - operator theory and Lie algebra theory. ... Read more

Customer Reviews (1)

Unique source of data
This book is the only place to find lots of theorems on exceptional Lie algebras (and therefore also on exceptional algebraic groups).For example, it is the only place I know of where it is proven that two27-dimensional exceptional Jordan algebras are isotopic if and only iftheir norm forms are similar.As Wallach said, this book also explains thedescription of the roots systems for the exceptional algebras found inJacobson's other book "Lie algebras".

The description of theD_4 (Spin_8) as a subalgebra of F_4 is quite beautiful. ... Read more

Book Description

Book Description
The volume is almost entirely composed of the research and expository papers by the participants of the International Workshop "Groups, Rings, Lie and Hopf Algebras", which was held at the Memorial University of Newfoundland, St. John's, NF, Canada. All four areas from the title of the workshop are covered. In addition, some chapters touch upon the topics, which belong to two or more areas at the same time.Audience: The readership targeted includes researchers, graduate and senior undergraduate students in mathematics and its applications. ... Read more